As shown in the figure, in the plane rectangular coordinate system, it is known that △AOB is an equilateral triangle with point A as its vertex.

The coordinate of is (0,4), point B is in the first quadrant, and point P is the moving point on the X axis. Connect AP, rotate △AOP counterclockwise around point A, so that the sides AO and AB overlap, and thus get △ABD.

(1) Find the analytical formula of straight line AB;

(2) When point P moves to point (0), find the length of DP at this time and the coordinates of point D;

(3) Whether there is a point P, so that the areas of △OPD are equal, and if there is, the coordinates of the point P satisfying the conditions are requested; If it does not exist, please explain why.

23. (This question 10)

As shown in figure 1, it is known that hyperbola and straight line intersect at a and b.

At two o'clock, point A is in the first quadrant. Try to answer the following questions:

(1) If the coordinate of point A is (4,2), the coordinate of point B is ▲; Ruodian a

The abscissa of is m, then the coordinate of point B can be expressed as ▲;

(2) As shown in Figure 2, draw a straight line L passing through the origin O and passing through the hyperbola.

P, q, point p is in the first quadrant.

① Explain that the quadrilateral APBQ must be a parallelogram;

② The abscissas of point A and point P are m and n respectively. Can quadrilateral APBQ be a rectangle?

Will it be a square? If possible, directly write the conditions that M and N should meet; If not.

Maybe. Please explain why.

Four, the topic (5 points)

Please note: this topic is entitled self-selected topic for candidates to choose. The scores of self-selected topics are included in the total score of this course, but the total score of the exam is the most 120.

25. For a quadratic function, if the function value is an integer when any integer is taken, then we call the image of the function an integer parabola (for example:).

(1) Please write the analytical formula of the parabola of the whole point with the absolute value of the quadratic term coefficient less than 1. (No proof required)

(2) Please explore: Is there an integral parabola whose absolute value is less than the quadratic coefficient? If it exists, please write an analytical formula of parabola; If it does not exist, please explain why.

As shown in figure 1, the vertices a and c of the right-angled trapezoidal OABC are on the positive and negative semi-axes of the Y axis, respectively. After passing through point B and point C, the straight line is translated, and the translated straight line intersects the axis of point D and the axis of point E. 。

(1) translate the straight line to the right, let the translation distance CD be (t 0), and the area swept by the straight line (the shaded part in the figure) be. The correlation function image is shown in Figure 2. OM is a line segment, MN is a part of a parabola, NQ is a ray, and the abscissa of n points is 4.

① Find the length of the trapezoid upper bottom AB and the area of the right-angled trapezoid OABC;

(2) When, find the resolution function of S;

(2) Under the condition of the problem (1), when the straight line moves to the left or right (including overlapping with the straight line BC), is there a point P on the straight line AB, which makes it an isosceles right triangle? If it exists, please directly write the coordinates of all points p that meet the conditions; If it does not exist, please explain why.

23. As shown in figure 1, the quadrilateral ABCD is a square, and G is a moving point on the edge of CD (point G does not coincide with C and D). Take CG as one side and make a square CEFG outside the square ABCD to connect BG and de. We explore the length relationship between line segment BG and line segment DE and the position relationship of straight line in the following figure:

(1)① guess the length relationship between line segment BG and line segment DE and the position relationship of straight line as shown in figure1;

② Rotate the square CEFG in figure 1 clockwise (or counterclockwise) at any angle around point C, and get the situation as shown in figures 2 and 3. Please observe and measure whether the conclusion drawn from diagram 1 is still valid, and choose Figure 2 to prove your judgment.

(2) The square in the original problem is changed into a rectangle (as shown in Figure 4-6), AB=a, BC=b, CE=ka, CG=kb (a b, k 0). Which conclusions are valid and which are not? If so, take Figure 5 as an example to briefly explain the reasons.

(3) In Figure 5 of the problem (2), connect,, and a=3, b=2, k=, and evaluate.